= Solution
Let $G$ be <Gibbs free energy>, $H$ <enthalpy> and $T$ <thermodynamic temperature> for a fixed amount and composition of material, with the <derivative> taken at constant <pressure>. Apply the <Gibbs–Helmholtz equation> to the standard molar reaction quantities, obtained by summing product minus reactant contributions with their stoichiometric coefficients:
$$
\frac{\mathrm d}{\mathrm dT}\left(\frac{\Delta_rG^\circ}{T}\right)=-\frac{\Delta_rH^\circ}{T^2}.
$$
Define $K(T)$ as the dimensionless thermodynamic <equilibrium constant> expressed using activities in specified standard states. Since $\Delta_rG^\circ=-RT\ln K$, where $R$ is the <molar gas constant>,
$$
-R\frac{\mathrm d\ln K}{\mathrm dT}=-\frac{\Delta_rH^\circ}{T^2},\qquad
\boxed{\frac{\mathrm d\ln K}{\mathrm dT}=\frac{\Delta_rH^\circ}{RT^2}.}
$$
If the reaction <enthalpy> is constant over the interval, <integration> gives
$$
\boxed{\ln\frac{K(T_2)}{K(T_1)}=-\frac{\Delta_rH^\circ}{R}\left(\frac1{T_2}-\frac1{T_1}\right).}
$$
This is the requested <Van't Hoff equation>. For an exothermic reaction, increasing <temperature> decreases $K$. Although the name “isochore” is often used in this context, the displayed <enthalpy> form follows from standard Gibbs energies and the stated constant-pressure Gibbs–Helmholtz identity; the analogous ideal-gas <concentration> form involves <internal energy> when the <temperature> dependence of standard <concentration> is retained.
Back to article page
